Isometric embedding of ${\ell }_1$ into Lipschitz-free spaces and ${\ell }_{\infty }$ into their duals

Abstract

We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of ${\ell }_{\infty }$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to ${\ell }_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~${\ell }_{\infty }$ into the dual" and "containing as good copy of~${\ell }_1$ as possible" in a general Banach space.